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Popular >

cot^2(x)+4cot(x)+3>0

Solution

cot^{2} (x) + 4 cot (x) + 3 > 0

Solution

πn < x < \frac{3 π}{4} + πn or arccot (- 3) + πn < x < π + πn

Interval Notation

(πn, \frac{3 π}{4} + πn) \cup (arccot (- 3) + πn, π + πn)

Decimal Notation

πn < x < 2.35619 \dots + πn or 2.81984 \dots + πn < x < 3.14159 \dots + πn

Solution steps

cot^{2} (x) + 4 cot (x) + 3 > 0

Let:

u = cot (x)

u^{2} + 4 u + 3 > 0

u^{2} + 4 u + 3 > 0 : u < - 3 or u > - 1

u^{2} + 4 u + 3 > 0

Factor

u^{2} + 4 u + 3 : (u + 1) (u + 3)

u^{2} + 4 u + 3

Break the expression into groups

u^{2} + 4 u + 3

Definition

Factors of

3 : 1, 3

3

Divisors (Factors)

Find the Prime factors of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Add 1

1

The factors of

3

1, 3

For every two factors such that

u * v = 3,

check if

u + v = 4

Check

u = 1, v = 3 : u * v = 3, u + v = 4 \Rightarrow

True

u = 1, v = 3

Group into

(a x^{2} + ux) + (vx + c)

(u^{2} + u) + (3 u + 3)

= (u^{2} + u) + (3 u + 3)

Factor out

u

from

u^{2} + u : u (u + 1)

u^{2} + u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= uu + u

Factor out common term

u

= u (u + 1)

Factor out

3

from

3 u + 3 : 3 (u + 1)

3 u + 3

Factor out common term

3

= 3 (u + 1)

= u (u + 1) + 3 (u + 1)

Factor out common term

u + 1

= (u + 1) (u + 3)

(u + 1) (u + 3) > 0

Identify the intervals

Find the signs of the factors of

(u + 1) (u + 3)

Find the signs of

u + 1

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

u + 1 < 0 : u < - 1

u + 1 < 0

Move

1

to the right side

u + 1 < 0

Subtract

1

from both sides

u + 1 - 1 < 0 - 1

Simplify

u < - 1

u < - 1

u + 1 > 0 : u > - 1

u + 1 > 0

Move

1

to the right side

u + 1 > 0

Subtract

1

from both sides

u + 1 - 1 > 0 - 1

Simplify

u > - 1

u > - 1

Find the signs of

u + 3

u + 3 = 0 : u = - 3

u + 3 = 0

Move

3

to the right side

u + 3 = 0

Subtract

3

from both sides

u + 3 - 3 = 0 - 3

Simplify

u = - 3

u = - 3

u + 3 < 0 : u < - 3

u + 3 < 0

Move

3

to the right side

u + 3 < 0

Subtract

3

from both sides

u + 3 - 3 < 0 - 3

Simplify

u < - 3

u < - 3

u + 3 > 0 : u > - 3

u + 3 > 0

Move

3

to the right side

u + 3 > 0

Subtract

3

from both sides

u + 3 - 3 > 0 - 3

Simplify

u > - 3

u > - 3

Summarize in a table:

u + 1 u + 3 (u + 1) (u + 3) u < - 3 - - + u = - 3 - 00 - 3 < u < - 1 - + - u = - 1 0 + 0 u > - 1 + + +

Identify the intervals that satisfy the required condition:

> 0

u < - 3 or u > - 1

u < - 3 or u > - 1

u < - 3 or u > - 1

Substitute back

u = cot (x)

cot (x) < - 3 or cot (x) > - 1

cot (x) < - 3 : arccot (- 3) + πn < x < π + πn

cot (x) < - 3

cot (x) < a

then

arccot (a) + πn < x < π + πn

arccot (- 3) + πn < x < π + πn

cot (x) > - 1 : πn < x < \frac{3 π}{4} + πn

cot (x) > - 1

cot (x) > a

then

πn < x < arccot (a) + πn

πn < x < arccot (- 1) + πn

Simplify

arccot (- 1) : \frac{3 π}{4}

arccot (- 1)

Use the following trivial identity:

arccot (- 1) = \frac{3 π}{4}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{3 π}{4}

πn < x < \frac{3 π}{4} + πn

Combine the intervals

arccot (- 3) + πn < x < π + πn or πn < x < \frac{3 π}{4} + πn

Merge Overlapping Intervals

πn < x < \frac{3 π}{4} + πn or arccot (- 3) + πn < x < π + πn

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